_f:=(k,n,X)->f[k]*product(X-x[_i],_i=0..k-1)*product(X-x[_i],_i=k+1..n)/
> (product(x[k]-x[_i],_i=0..k-1)*product(x[k]-x[_i],_i=k+1..n));
_f := (k, n, X) ->
/ k - 1 \ / n \
| --------' | | --------' |
|' | | | |' | | |
f[k] | | | (X - x[_i])| | | | (X - x[_i])|
| | | | | | | |
| | | | | | | |
\ _i = 0 / \_i = k + 1 /
-------------------------------------------------------
/ k - 1 \ / n \
| --------' | | --------' |
|' | | | |' | | |
| | | (x[k] - x[_i])| | | | (x[k] - x[_i])|
| | | | | | | |
| | | | | | | |
\ _i = 0 / \_i = k + 1 /
>
> F:=(y,n)->Sum(_f(j,n,y),j=(0..n));
_N
-----
\
F := (y, _N) -> ) _f(j, _N, y)
/
-----
j = 0
Here :
> F(x[3],3);
>
3
-----
\
) 0
/
-----
j = 0
But ,
> _f(0,3,x[3])+_f(1,3,x[3])+_f(2,3,x[3])+_f(3,3,x[3]);
f[3]
> F(x[3],4);
4 / j - 1 \ / 4 \
----- | --------' | | --------' |
\ |' | | | |' | | |
) f[j] | | | (x[3] - x[_i])| | | | (x[3] - x[_i])|
/ | | | | | | | |
----- | | | | | | | |
j = 0 \ _i = 0 / \_i = j + 1 /
// j - 1 \ / 4 \
|| --------' | | --------' |
/ ||' | | | |' | | |
/ || | | (x[j] - x[_i])| | | | (x[j] - x[_i])|
/ || | | | | | | |
|| | | | | | | |
\\ _i = 0 / \_i = j + 1 /
\
|
|
|
|
|
/
